Analysis of functions of a single variable via multivariable calculus?

[imwhatim]

Took calculus of a single variable almost a decade ago where every theorem had to be accepted without proof. Can I fill these gaps by studying a rigorous multivariable/vector analysis book? My justification for this is that R^1 is just a special case of R^n. Or am I looking at this the wrong way and proving things in R^n requires results from R^1?

 

[fresh_42]

You are basically right, but I think there are theorems that do not occur in vector calculus. Others may be proven by a reduction to the one-dimensional case which won't be repeated. Others might use the one-dimensional case as an induction basis. Real analysis isn't properly included in vector calculus.

In the end, it depends on your goals. E.g. you could consider reading a calculus textbook as warm-up for vector calculus. If it is a matter of sources, then you could look for lecture notes on the internet: google "Calculus I + pdf".

 

[imwhatim]

If it is a matter of sources, then you could look for lecture notes on the internet: google "Calculus I + pdf".

It's mostly a matter of being efficient with time, I didn't want to go over cal 1/2 again. I'm making a return to school and the curriculum suggests multivariable calculus. If I pick a rigorous book to go along with the course, I figured I could kill two birds with one stone.

 

[fresh_42]

It's mostly a matter of being efficient with time, I didn't want to go over cal 1/2 again. I'm making a return to school and the curriculum suggests multivariable calculus. If I pick a rigorous book to go along with the course, I figured I could kill two birds with one stone.

There is a reason that Calculus I exists. If it were covered by multivariate calculus, then it wouldn't be taught at universities. Rolle and IVT are important theorems.

 

[WWGD]

Still, issues like differentiability in $\mathbb R^n$ are more " twisted" than in $\mathbb R$ itself, as limits must agree along all possible directions. Even if all partials exist for your function with n arguments, the function may not be differentiable.

 

[fresh_42]

Still, issues like differentiability in $\mathbb R^n$ are more " twisted" than in $\mathbb R$ itself, as limits must agree along all possible directions. Even if all partials exist for your function with n arguments, the function may not be differentiable.

Once again an argument for my favored version of the differentiability definition: Weierstraß! One formula fits all.

 

[fresh_42]

There is even another level of this argument! Forget about multivariate calculus. Let's talk about fiber bundles instead!

 

[Frabjous]

Check out one of the more theoretical calculus books
Courant
Spivak
Apostal
It is a different experience, so it will not be as repetitive as you think.

 

[WWGD]

There is even another level of this argument! Forget about multivariate calculus. Let's talk about fiber bundles instead!

How does that fit in?

 

[fresh_42]

How does that fit in?

If you drop single-variate calculus for multivariate calculus, you could as well start with fiber bundles, sections, and differential forms to spare multivariate calculus. There is always a generalization.

That would be a nice thread / insight / or just fun: take rudimentary theorems like Rolle or IVT and rephrase them in terms of tangent bundles, sections, and differential forms, i.e. Graßman algebras. I mean they did the same thing with the fundamental theorem of calculus. It appears in so many different ways, and every version has a name: Stokes, Divergence, Gauß-Bonnet, Cauchy, and probably some more. All are about the question of how much information in which situation is already provided on the boundaries. I've read these days that even the seven bridges of Königsberg can be seen from the perspective of FTC as an early version. The association chain was: Königsberg > Euler characteristic > Gauß-Bonnet > triangularizations > Stokes > FTC.

 

[WWGD]

I see, so maps from $\mathbb R^n \rightarrow \mathbb R^m$ i.e., Vector Fields, viewed as sections of bundles, boundary conditions related to cconditions in the interior, etc?